LMFDB - Embedded newform 8002.2.a.d.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.8962916974$
Analytic rank:	$1$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.96349 q^{3} +1.00000 q^{4} +0.136598 q^{5} -1.96349 q^{6} -0.645165 q^{7} +1.00000 q^{8} +0.855285 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.96349 q^{3} +1.00000 q^{4} +0.136598 q^{5} -1.96349 q^{6} -0.645165 q^{7} +1.00000 q^{8} +0.855285 q^{9} +0.136598 q^{10} +5.43439 q^{11} -1.96349 q^{12} +2.62554 q^{13} -0.645165 q^{14} -0.268208 q^{15} +1.00000 q^{16} +0.137394 q^{17} +0.855285 q^{18} -3.70721 q^{19} +0.136598 q^{20} +1.26677 q^{21} +5.43439 q^{22} -7.68849 q^{23} -1.96349 q^{24} -4.98134 q^{25} +2.62554 q^{26} +4.21112 q^{27} -0.645165 q^{28} -5.85445 q^{29} -0.268208 q^{30} +7.74561 q^{31} +1.00000 q^{32} -10.6704 q^{33} +0.137394 q^{34} -0.0881280 q^{35} +0.855285 q^{36} -8.38418 q^{37} -3.70721 q^{38} -5.15522 q^{39} +0.136598 q^{40} +7.12279 q^{41} +1.26677 q^{42} -9.80190 q^{43} +5.43439 q^{44} +0.116830 q^{45} -7.68849 q^{46} -7.62943 q^{47} -1.96349 q^{48} -6.58376 q^{49} -4.98134 q^{50} -0.269772 q^{51} +2.62554 q^{52} +6.45470 q^{53} +4.21112 q^{54} +0.742324 q^{55} -0.645165 q^{56} +7.27906 q^{57} -5.85445 q^{58} +10.6054 q^{59} -0.268208 q^{60} +10.8817 q^{61} +7.74561 q^{62} -0.551800 q^{63} +1.00000 q^{64} +0.358642 q^{65} -10.6704 q^{66} -10.7922 q^{67} +0.137394 q^{68} +15.0963 q^{69} -0.0881280 q^{70} +7.43766 q^{71} +0.855285 q^{72} -2.53878 q^{73} -8.38418 q^{74} +9.78080 q^{75} -3.70721 q^{76} -3.50608 q^{77} -5.15522 q^{78} +1.30045 q^{79} +0.136598 q^{80} -10.8343 q^{81} +7.12279 q^{82} -17.2760 q^{83} +1.26677 q^{84} +0.0187677 q^{85} -9.80190 q^{86} +11.4951 q^{87} +5.43439 q^{88} -10.7479 q^{89} +0.116830 q^{90} -1.69391 q^{91} -7.68849 q^{92} -15.2084 q^{93} -7.62943 q^{94} -0.506395 q^{95} -1.96349 q^{96} -1.49348 q^{97} -6.58376 q^{98} +4.64795 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) $ 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.96349	−1.13362	−0.566810	−	0.823848i	$-0.691823\pi$
$3$	−1.96349	−1.13362	−0.566810	+	0.823848i	$0.691823\pi$
$4$	1.00000	0.500000
$5$	0.136598	0.0610883	0.0305441	−	0.999533i	$-0.490276\pi$
$5$	0.136598	0.0610883	0.0305441	+	0.999533i	$0.490276\pi$
$6$	−1.96349	−0.801591
$7$	−0.645165	−0.243850	−0.121925	−	0.992539i	$-0.538907\pi$
$7$	−0.645165	−0.243850	−0.121925	+	0.992539i	$0.538907\pi$
$8$	1.00000	0.353553
$9$	0.855285	0.285095
$10$	0.136598	0.0431959
$11$	5.43439	1.63853	0.819265	−	0.573415i	$-0.194382\pi$
$11$	5.43439	1.63853	0.819265	+	0.573415i	$0.194382\pi$
$12$	−1.96349	−0.566810
$13$	2.62554	0.728194	0.364097	−	0.931361i	$-0.381378\pi$
$13$	2.62554	0.728194	0.364097	+	0.931361i	$0.381378\pi$
$14$	−0.645165	−0.172428
$15$	−0.268208	−0.0692509
$16$	1.00000	0.250000
$17$	0.137394	0.0333230	0.0166615	−	0.999861i	$-0.494696\pi$
$17$	0.137394	0.0333230	0.0166615	+	0.999861i	$0.494696\pi$
$18$	0.855285	0.201593
$19$	−3.70721	−0.850492	−0.425246	−	0.905078i	$-0.639812\pi$
$19$	−3.70721	−0.850492	−0.425246	+	0.905078i	$0.639812\pi$
$20$	0.136598	0.0305441
$21$	1.26677	0.276433
$22$	5.43439	1.15862
$23$	−7.68849	−1.60316	−0.801580	−	0.597887i	$-0.796007\pi$
$23$	−7.68849	−1.60316	−0.801580	+	0.597887i	$0.796007\pi$
$24$	−1.96349	−0.400795
$25$	−4.98134	−0.996268
$26$	2.62554	0.514911
$27$	4.21112	0.810431
$28$	−0.645165	−0.121925
$29$	−5.85445	−1.08714	−0.543572	−	0.839362i	$-0.682929\pi$
$29$	−5.85445	−1.08714	−0.543572	+	0.839362i	$0.682929\pi$
$30$	−0.268208	−0.0489678
$31$	7.74561	1.39115	0.695576	−	0.718452i	$-0.255149\pi$
$31$	7.74561	1.39115	0.695576	+	0.718452i	$0.255149\pi$
$32$	1.00000	0.176777
$33$	−10.6704	−1.85747
$34$	0.137394	0.0235629
$35$	−0.0881280	−0.0148963
$36$	0.855285	0.142547
$37$	−8.38418	−1.37835	−0.689175	−	0.724595i	$-0.742027\pi$
$37$	−8.38418	−1.37835	−0.689175	+	0.724595i	$0.742027\pi$
$38$	−3.70721	−0.601389
$39$	−5.15522	−0.825496
$40$	0.136598	0.0215980
$41$	7.12279	1.11239	0.556197	−	0.831051i	$-0.312260\pi$
$41$	7.12279	1.11239	0.556197	+	0.831051i	$0.312260\pi$
$42$	1.26677	0.195468
$43$	−9.80190	−1.49478	−0.747388	−	0.664388i	$-0.768692\pi$
$43$	−9.80190	−1.49478	−0.747388	+	0.664388i	$0.768692\pi$
$44$	5.43439	0.819265
$45$	0.116830	0.0174160
$46$	−7.68849	−1.13361
$47$	−7.62943	−1.11287	−0.556433	−	0.830892i	$-0.687831\pi$
$47$	−7.62943	−1.11287	−0.556433	+	0.830892i	$0.687831\pi$
$48$	−1.96349	−0.283405
$49$	−6.58376	−0.940537
$50$	−4.98134	−0.704468
$51$	−0.269772	−0.0377757
$52$	2.62554	0.364097
$53$	6.45470	0.886621	0.443311	−	0.896368i	$-0.353804\pi$
$53$	6.45470	0.886621	0.443311	+	0.896368i	$0.353804\pi$
$54$	4.21112	0.573061
$55$	0.742324	0.100095
$56$	−0.645165	−0.0862138
$57$	7.27906	0.964135
$58$	−5.85445	−0.768727
$59$	10.6054	1.38070	0.690351	−	0.723475i	$-0.257456\pi$
$59$	10.6054	1.38070	0.690351	+	0.723475i	$0.257456\pi$
$60$	−0.268208	−0.0346254
$61$	10.8817	1.39325	0.696627	−	0.717434i	$-0.254683\pi$
$61$	10.8817	1.39325	0.696627	+	0.717434i	$0.254683\pi$
$62$	7.74561	0.983693
$63$	−0.551800	−0.0695203
$64$	1.00000	0.125000
$65$	0.358642	0.0444841
$66$	−10.6704	−1.31343
$67$	−10.7922	−1.31848	−0.659240	−	0.751933i	$-0.729122\pi$
$67$	−10.7922	−1.31848	−0.659240	+	0.751933i	$0.729122\pi$
$68$	0.137394	0.0166615
$69$	15.0963	1.81738
$70$	−0.0881280	−0.0105333
$71$	7.43766	0.882687	0.441344	−	0.897338i	$-0.354502\pi$
$71$	7.43766	0.882687	0.441344	+	0.897338i	$0.354502\pi$
$72$	0.855285	0.100796
$73$	−2.53878	−0.297141	−0.148571	−	0.988902i	$-0.547467\pi$
$73$	−2.53878	−0.297141	−0.148571	+	0.988902i	$0.547467\pi$
$74$	−8.38418	−0.974641
$75$	9.78080	1.12939
$76$	−3.70721	−0.425246
$77$	−3.50608	−0.399555
$78$	−5.15522	−0.583714
$79$	1.30045	0.146312	0.0731559	−	0.997321i	$-0.476693\pi$
$79$	1.30045	0.146312	0.0731559	+	0.997321i	$0.476693\pi$
$80$	0.136598	0.0152721
$81$	−10.8343	−1.20382
$82$	7.12279	0.786581
$83$	−17.2760	−1.89629	−0.948146	−	0.317834i	$-0.897044\pi$
$83$	−17.2760	−1.89629	−0.948146	+	0.317834i	$0.897044\pi$
$84$	1.26677	0.138216
$85$	0.0187677	0.00203565
$86$	−9.80190	−1.05697
$87$	11.4951	1.23241
$88$	5.43439	0.579308
$89$	−10.7479	−1.13927	−0.569635	−	0.821898i	$-0.692916\pi$
$89$	−10.7479	−1.13927	−0.569635	+	0.821898i	$0.692916\pi$
$90$	0.116830	0.0123149
$91$	−1.69391	−0.177570
$92$	−7.68849	−0.801580
$93$	−15.2084	−1.57704
$94$	−7.62943	−0.786916
$95$	−0.506395	−0.0519551
$96$	−1.96349	−0.200398
$97$	−1.49348	−0.151640	−0.0758198	−	0.997122i	$-0.524157\pi$
$97$	−1.49348	−0.151640	−0.0758198	+	0.997122i	$0.524157\pi$
$98$	−6.58376	−0.665060
$99$	4.64795	0.467137
$100$	−4.98134	−0.498134
$101$	−0.169838	−0.0168995	−0.00844974	−	0.999964i	$-0.502690\pi$
$101$	−0.169838	−0.0168995	−0.00844974	+	0.999964i	$0.502690\pi$
$102$	−0.269772	−0.0267114
$103$	−8.97453	−0.884286	−0.442143	−	0.896944i	$-0.645782\pi$
$103$	−8.97453	−0.884286	−0.442143	+	0.896944i	$0.645782\pi$
$104$	2.62554	0.257456
$105$	0.173038	0.0168868
$106$	6.45470	0.626936
$107$	16.3362	1.57928	0.789638	−	0.613573i	$-0.210268\pi$
$107$	16.3362	1.57928	0.789638	+	0.613573i	$0.210268\pi$
$108$	4.21112	0.405215
$109$	−2.81281	−0.269418	−0.134709	−	0.990885i	$-0.543010\pi$
$109$	−2.81281	−0.269418	−0.134709	+	0.990885i	$0.543010\pi$
$110$	0.742324	0.0707778
$111$	16.4622	1.56253
$112$	−0.645165	−0.0609624
$113$	2.96629	0.279045	0.139523	−	0.990219i	$-0.455443\pi$
$113$	2.96629	0.279045	0.139523	+	0.990219i	$0.455443\pi$
$114$	7.27906	0.681746
$115$	−1.05023	−0.0979343
$116$	−5.85445	−0.543572
$117$	2.24559	0.207604
$118$	10.6054	0.976303
$119$	−0.0886421	−0.00812581
$120$	−0.268208	−0.0244839
$121$	18.5326	1.68478
$122$	10.8817	0.985179
$123$	−13.9855	−1.26103
$124$	7.74561	0.695576
$125$	−1.36343	−0.121949
$126$	−0.551800	−0.0491583
$127$	8.58178	0.761509	0.380755	−	0.924676i	$-0.375664\pi$
$127$	8.58178	0.761509	0.380755	+	0.924676i	$0.375664\pi$
$128$	1.00000	0.0883883
$129$	19.2459	1.69451
$130$	0.358642	0.0314550
$131$	18.2274	1.59254	0.796269	−	0.604943i	$-0.206804\pi$
$131$	18.2274	1.59254	0.796269	+	0.604943i	$0.206804\pi$
$132$	−10.6704	−0.928735
$133$	2.39176	0.207392
$134$	−10.7922	−0.932306
$135$	0.575229	0.0495078
$136$	0.137394	0.0117815
$137$	2.28829	0.195502	0.0977509	−	0.995211i	$-0.468835\pi$
$137$	2.28829	0.195502	0.0977509	+	0.995211i	$0.468835\pi$
$138$	15.0963	1.28508
$139$	−4.39175	−0.372503	−0.186252	−	0.982502i	$-0.559634\pi$
$139$	−4.39175	−0.372503	−0.186252	+	0.982502i	$0.559634\pi$
$140$	−0.0881280	−0.00744817
$141$	14.9803	1.26157
$142$	7.43766	0.624154
$143$	14.2682	1.19317
$144$	0.855285	0.0712737
$145$	−0.799704	−0.0664118
$146$	−2.53878	−0.210111
$147$	12.9271	1.06621
$148$	−8.38418	−0.689175
$149$	16.2633	1.33234	0.666172	−	0.745798i	$-0.267932\pi$
$149$	16.2633	1.33234	0.666172	+	0.745798i	$0.267932\pi$
$150$	9.78080	0.798599
$151$	−6.30411	−0.513021	−0.256511	−	0.966541i	$-0.582573\pi$
$151$	−6.30411	−0.513021	−0.256511	+	0.966541i	$0.582573\pi$
$152$	−3.70721	−0.300694
$153$	0.117511	0.00950023
$154$	−3.50608	−0.282528
$155$	1.05803	0.0849831
$156$	−5.15522	−0.412748
$157$	8.71210	0.695301	0.347651	−	0.937624i	$-0.386980\pi$
$157$	8.71210	0.695301	0.347651	+	0.937624i	$0.386980\pi$
$158$	1.30045	0.103458
$159$	−12.6737	−1.00509
$160$	0.136598	0.0107990
$161$	4.96035	0.390930
$162$	−10.8343	−0.851226
$163$	−13.7071	−1.07362	−0.536810	−	0.843703i	$-0.680371\pi$
$163$	−13.7071	−1.07362	−0.536810	+	0.843703i	$0.680371\pi$
$164$	7.12279	0.556197
$165$	−1.45754	−0.113470
$166$	−17.2760	−1.34088
$167$	−22.8476	−1.76800	−0.883999	−	0.467489i	$-0.845159\pi$
$167$	−22.8476	−1.76800	−0.883999	+	0.467489i	$0.845159\pi$
$168$	1.26677	0.0977338
$169$	−6.10653	−0.469733
$170$	0.0187677	0.00143942
$171$	−3.17072	−0.242471
$172$	−9.80190	−0.747388
$173$	−13.7361	−1.04433	−0.522167	−	0.852843i	$-0.674876\pi$
$173$	−13.7361	−1.04433	−0.522167	+	0.852843i	$0.674876\pi$
$174$	11.4951	0.871445
$175$	3.21379	0.242940
$176$	5.43439	0.409632
$177$	−20.8235	−1.56519
$178$	−10.7479	−0.805586
$179$	0.228136	0.0170517	0.00852585	−	0.999964i	$-0.497286\pi$
$179$	0.228136	0.0170517	0.00852585	+	0.999964i	$0.497286\pi$
$180$	0.116830	0.00870798
$181$	−23.6984	−1.76149	−0.880745	−	0.473592i	$-0.842957\pi$
$181$	−23.6984	−1.76149	−0.880745	+	0.473592i	$0.842957\pi$
$182$	−1.69391	−0.125561
$183$	−21.3660	−1.57942
$184$	−7.68849	−0.566803
$185$	−1.14526	−0.0842010
$186$	−15.2084	−1.11513
$187$	0.746655	0.0546008
$188$	−7.62943	−0.556433
$189$	−2.71687	−0.197623
$190$	−0.506395	−0.0367378
$191$	−19.4428	−1.40683	−0.703417	−	0.710777i	$-0.748343\pi$
$191$	−19.4428	−1.40683	−0.703417	+	0.710777i	$0.748343\pi$
$192$	−1.96349	−0.141703
$193$	−9.37529	−0.674848	−0.337424	−	0.941353i	$-0.609556\pi$
$193$	−9.37529	−0.674848	−0.337424	+	0.941353i	$0.609556\pi$
$194$	−1.49348	−0.107225
$195$	−0.704190	−0.0504281
$196$	−6.58376	−0.470269
$197$	−14.8784	−1.06004	−0.530021	−	0.847984i	$-0.677816\pi$
$197$	−14.8784	−1.06004	−0.530021	+	0.847984i	$0.677816\pi$
$198$	4.64795	0.330315
$199$	17.5989	1.24756	0.623778	−	0.781602i	$-0.285597\pi$
$199$	17.5989	1.24756	0.623778	+	0.781602i	$0.285597\pi$
$200$	−4.98134	−0.352234
$201$	21.1904	1.49466
$202$	−0.169838	−0.0119497
$203$	3.77709	0.265100
$204$	−0.269772	−0.0188878
$205$	0.972956	0.0679542
$206$	−8.97453	−0.625285
$207$	−6.57585	−0.457053
$208$	2.62554	0.182049
$209$	−20.1464	−1.39356
$210$	0.173038	0.0119408
$211$	24.7428	1.70336	0.851682	−	0.524059i	$-0.175583\pi$
$211$	24.7428	1.70336	0.851682	+	0.524059i	$0.175583\pi$
$212$	6.45470	0.443311
$213$	−14.6037	−1.00063
$214$	16.3362	1.11672
$215$	−1.33891	−0.0913132
$216$	4.21112	0.286531
$217$	−4.99720	−0.339232
$218$	−2.81281	−0.190507
$219$	4.98486	0.336846
$220$	0.742324	0.0500475
$221$	0.360735	0.0242656
$222$	16.4622	1.10487
$223$	−11.6833	−0.782374	−0.391187	−	0.920311i	$-0.627935\pi$
$223$	−11.6833	−0.782374	−0.391187	+	0.920311i	$0.627935\pi$
$224$	−0.645165	−0.0431069
$225$	−4.26046	−0.284031
$226$	2.96629	0.197315
$227$	14.3672	0.953582	0.476791	−	0.879017i	$-0.341800\pi$
$227$	14.3672	0.953582	0.476791	+	0.879017i	$0.341800\pi$
$228$	7.27906	0.482067
$229$	−9.27281	−0.612764	−0.306382	−	0.951909i	$-0.599119\pi$
$229$	−9.27281	−0.612764	−0.306382	+	0.951909i	$0.599119\pi$
$230$	−1.05023	−0.0692500
$231$	6.88414	0.452943
$232$	−5.85445	−0.384364
$233$	28.5096	1.86772	0.933861	−	0.357635i	$-0.116417\pi$
$233$	28.5096	1.86772	0.933861	+	0.357635i	$0.116417\pi$
$234$	2.24559	0.146799
$235$	−1.04216	−0.0679831
$236$	10.6054	0.690351
$237$	−2.55341	−0.165862
$238$	−0.0886421	−0.00574581
$239$	−1.92886	−0.124767	−0.0623837	−	0.998052i	$-0.519870\pi$
$239$	−1.92886	−0.124767	−0.0623837	+	0.998052i	$0.519870\pi$
$240$	−0.268208	−0.0173127
$241$	−17.7798	−1.14530	−0.572650	−	0.819800i	$-0.694084\pi$
$241$	−17.7798	−1.14530	−0.572650	+	0.819800i	$0.694084\pi$
$242$	18.5326	1.19132
$243$	8.63973	0.554239
$244$	10.8817	0.696627
$245$	−0.899325	−0.0574558
$246$	−13.9855	−0.891684
$247$	−9.73343	−0.619323
$248$	7.74561	0.491847
$249$	33.9213	2.14968
$250$	−1.36343	−0.0862307
$251$	−21.1878	−1.33736	−0.668680	−	0.743551i	$-0.733140\pi$
$251$	−21.1878	−1.33736	−0.668680	+	0.743551i	$0.733140\pi$
$252$	−0.551800	−0.0347601
$253$	−41.7822	−2.62683
$254$	8.58178	0.538469
$255$	−0.0368502	−0.00230765
$256$	1.00000	0.0625000
$257$	−27.9902	−1.74598	−0.872990	−	0.487738i	$-0.837822\pi$
$257$	−27.9902	−1.74598	−0.872990	+	0.487738i	$0.837822\pi$
$258$	19.2459	1.19820
$259$	5.40918	0.336110
$260$	0.358642	0.0222421
$261$	−5.00722	−0.309939
$262$	18.2274	1.12609
$263$	−23.4759	−1.44758	−0.723792	−	0.690018i	$-0.757602\pi$
$263$	−23.4759	−1.44758	−0.723792	+	0.690018i	$0.757602\pi$
$264$	−10.6704	−0.656715
$265$	0.881696	0.0541621
$266$	2.39176	0.146648
$267$	21.1033	1.29150
$268$	−10.7922	−0.659240
$269$	−19.9201	−1.21455	−0.607275	−	0.794492i	$-0.707737\pi$
$269$	−19.9201	−1.21455	−0.607275	+	0.794492i	$0.707737\pi$
$270$	0.575229	0.0350073
$271$	−12.7918	−0.777046	−0.388523	−	0.921439i	$-0.627015\pi$
$271$	−12.7918	−0.777046	−0.388523	+	0.921439i	$0.627015\pi$
$272$	0.137394	0.00833076
$273$	3.32597	0.201297
$274$	2.28829	0.138241
$275$	−27.0705	−1.63242
$276$	15.0963	0.908688
$277$	18.8469	1.13240	0.566201	−	0.824267i	$-0.308413\pi$
$277$	18.8469	1.13240	0.566201	+	0.824267i	$0.308413\pi$
$278$	−4.39175	−0.263400
$279$	6.62470	0.396611
$280$	−0.0881280	−0.00526665
$281$	−12.9456	−0.772268	−0.386134	−	0.922443i	$-0.626190\pi$
$281$	−12.9456	−0.772268	−0.386134	+	0.922443i	$0.626190\pi$
$282$	14.9803	0.892063
$283$	−17.4489	−1.03723	−0.518614	−	0.855009i	$-0.673552\pi$
$283$	−17.4489	−1.03723	−0.518614	+	0.855009i	$0.673552\pi$
$284$	7.43766	0.441344
$285$	0.994301	0.0588973
$286$	14.2682	0.843697
$287$	−4.59538	−0.271257
$288$	0.855285	0.0503981
$289$	−16.9811	−0.998890
$290$	−0.799704	−0.0469602
$291$	2.93242	0.171902
$292$	−2.53878	−0.148571
$293$	−12.8542	−0.750950	−0.375475	−	0.926833i	$-0.622520\pi$
$293$	−12.8542	−0.750950	−0.375475	+	0.926833i	$0.622520\pi$
$294$	12.9271	0.753926
$295$	1.44867	0.0843447
$296$	−8.38418	−0.487320
$297$	22.8849	1.32792
$298$	16.2633	0.942109
$299$	−20.1865	−1.16741
$300$	9.78080	0.564695
$301$	6.32385	0.364500
$302$	−6.30411	−0.362761
$303$	0.333474	0.0191576
$304$	−3.70721	−0.212623
$305$	1.48641	0.0851114
$306$	0.117511	0.00671768
$307$	25.2718	1.44234	0.721168	−	0.692760i	$-0.243606\pi$
$307$	25.2718	1.44234	0.721168	+	0.692760i	$0.243606\pi$
$308$	−3.50608	−0.199777
$309$	17.6214	1.00244
$310$	1.05803	0.0600921
$311$	8.91856	0.505725	0.252862	−	0.967502i	$-0.418628\pi$
$311$	8.91856	0.505725	0.252862	+	0.967502i	$0.418628\pi$
$312$	−5.15522	−0.291857
$313$	−28.9112	−1.63416	−0.817079	−	0.576526i	$-0.804408\pi$
$313$	−28.9112	−1.63416	−0.817079	+	0.576526i	$0.804408\pi$
$314$	8.71210	0.491652
$315$	−0.0753745	−0.00424687
$316$	1.30045	0.0731559
$317$	−14.6613	−0.823458	−0.411729	−	0.911306i	$-0.635075\pi$
$317$	−14.6613	−0.823458	−0.411729	+	0.911306i	$0.635075\pi$
$318$	−12.6737	−0.710707
$319$	−31.8154	−1.78132
$320$	0.136598	0.00763603
$321$	−32.0758	−1.79030
$322$	4.96035	0.276429
$323$	−0.509350	−0.0283410
$324$	−10.8343	−0.601908
$325$	−13.0787	−0.725477
$326$	−13.7071	−0.759165
$327$	5.52291	0.305418
$328$	7.12279	0.393290
$329$	4.92224	0.271372
$330$	−1.45754	−0.0802352
$331$	6.90707	0.379647	0.189824	−	0.981818i	$-0.439208\pi$
$331$	6.90707	0.379647	0.189824	+	0.981818i	$0.439208\pi$
$332$	−17.2760	−0.948146
$333$	−7.17086	−0.392961
$334$	−22.8476	−1.25016
$335$	−1.47419	−0.0805436
$336$	1.26677	0.0691082
$337$	18.4880	1.00711	0.503554	−	0.863964i	$-0.332026\pi$
$337$	18.4880	1.00711	0.503554	+	0.863964i	$0.332026\pi$
$338$	−6.10653	−0.332151
$339$	−5.82428	−0.316331
$340$	0.0187677	0.00101782
$341$	42.0927	2.27945
$342$	−3.17072	−0.171453
$343$	8.76377	0.473199
$344$	−9.80190	−0.528483
$345$	2.06211	0.111020
$346$	−13.7361	−0.738455
$347$	8.90909	0.478265	0.239132	−	0.970987i	$-0.423137\pi$
$347$	8.90909	0.478265	0.239132	+	0.970987i	$0.423137\pi$
$348$	11.4951	0.616205
$349$	6.31107	0.337824	0.168912	−	0.985631i	$-0.445975\pi$
$349$	6.31107	0.337824	0.168912	+	0.985631i	$0.445975\pi$
$350$	3.21379	0.171784
$351$	11.0565	0.590151
$352$	5.43439	0.289654
$353$	3.99399	0.212579	0.106289	−	0.994335i	$-0.466103\pi$
$353$	3.99399	0.212579	0.106289	+	0.994335i	$0.466103\pi$
$354$	−20.8235	−1.10676
$355$	1.01597	0.0539218
$356$	−10.7479	−0.569635
$357$	0.174048	0.00921158
$358$	0.228136	0.0120574
$359$	21.0913	1.11316	0.556579	−	0.830795i	$-0.312114\pi$
$359$	21.0913	1.11316	0.556579	+	0.830795i	$0.312114\pi$
$360$	0.116830	0.00615747
$361$	−5.25661	−0.276664
$362$	−23.6984	−1.24556
$363$	−36.3885	−1.90990
$364$	−1.69391	−0.0887849
$365$	−0.346791	−0.0181519
$366$	−21.3660	−1.11682
$367$	7.10645	0.370954	0.185477	−	0.982649i	$-0.440617\pi$
$367$	7.10645	0.370954	0.185477	+	0.982649i	$0.440617\pi$
$368$	−7.68849	−0.400790
$369$	6.09201	0.317138
$370$	−1.14526	−0.0595391
$371$	−4.16435	−0.216202
$372$	−15.2084	−0.788519
$373$	−20.1340	−1.04250	−0.521249	−	0.853404i	$-0.674534\pi$
$373$	−20.1340	−1.04250	−0.521249	+	0.853404i	$0.674534\pi$
$374$	0.746655	0.0386086
$375$	2.67707	0.138243
$376$	−7.62943	−0.393458
$377$	−15.3711	−0.791652
$378$	−2.71687	−0.139741
$379$	−15.6170	−0.802190	−0.401095	−	0.916037i	$-0.631370\pi$
$379$	−15.6170	−0.802190	−0.401095	+	0.916037i	$0.631370\pi$
$380$	−0.506395	−0.0259775
$381$	−16.8502	−0.863263
$382$	−19.4428	−0.994782
$383$	−23.6516	−1.20854	−0.604270	−	0.796779i	$-0.706535\pi$
$383$	−23.6516	−1.20854	−0.604270	+	0.796779i	$0.706535\pi$
$384$	−1.96349	−0.100199
$385$	−0.478922	−0.0244081
$386$	−9.37529	−0.477190
$387$	−8.38341	−0.426153
$388$	−1.49348	−0.0758198
$389$	−5.10491	−0.258829	−0.129415	−	0.991591i	$-0.541310\pi$
$389$	−5.10491	−0.258829	−0.129415	+	0.991591i	$0.541310\pi$
$390$	−0.704190	−0.0356581
$391$	−1.05636	−0.0534222
$392$	−6.58376	−0.332530
$393$	−35.7893	−1.80533
$394$	−14.8784	−0.749564
$395$	0.177638	0.00893793
$396$	4.64795	0.233568
$397$	39.6218	1.98856	0.994281	−	0.106792i	$-0.0340580\pi$
$397$	39.6218	1.98856	0.994281	+	0.106792i	$0.0340580\pi$
$398$	17.5989	0.882155
$399$	−4.69620	−0.235104
$400$	−4.98134	−0.249067
$401$	−25.0412	−1.25050	−0.625248	−	0.780426i	$-0.715002\pi$
$401$	−25.0412	−1.25050	−0.625248	+	0.780426i	$0.715002\pi$
$402$	21.1904	1.05688
$403$	20.3364	1.01303
$404$	−0.169838	−0.00844974
$405$	−1.47994	−0.0735390
$406$	3.77709	0.187454
$407$	−45.5629	−2.25847
$408$	−0.269772	−0.0133557
$409$	30.8092	1.52342	0.761708	−	0.647920i	$-0.224361\pi$
$409$	30.8092	1.52342	0.761708	+	0.647920i	$0.224361\pi$
$410$	0.972956	0.0480508
$411$	−4.49303	−0.221625
$412$	−8.97453	−0.442143
$413$	−6.84222	−0.336683
$414$	−6.57585	−0.323185
$415$	−2.35986	−0.115841
$416$	2.62554	0.128728
$417$	8.62314	0.422277
$418$	−20.1464	−0.985393
$419$	23.2766	1.13714	0.568568	−	0.822636i	$-0.307498\pi$
$419$	23.2766	1.13714	0.568568	+	0.822636i	$0.307498\pi$
$420$	0.173038	0.00844340
$421$	−16.6580	−0.811859	−0.405930	−	0.913904i	$-0.633052\pi$
$421$	−16.6580	−0.811859	−0.405930	+	0.913904i	$0.633052\pi$
$422$	24.7428	1.20446
$423$	−6.52533	−0.317273
$424$	6.45470	0.313468
$425$	−0.684408	−0.0331987
$426$	−14.6037	−0.707554
$427$	−7.02047	−0.339744
$428$	16.3362	0.789638
$429$	−28.0155	−1.35260
$430$	−1.33891	−0.0645682
$431$	3.10680	0.149649	0.0748246	−	0.997197i	$-0.476160\pi$
$431$	3.10680	0.149649	0.0748246	+	0.997197i	$0.476160\pi$
$432$	4.21112	0.202608
$433$	−17.9951	−0.864789	−0.432395	−	0.901684i	$-0.642331\pi$
$433$	−17.9951	−0.864789	−0.432395	+	0.901684i	$0.642331\pi$
$434$	−4.99720	−0.239873
$435$	1.57021	0.0752857
$436$	−2.81281	−0.134709
$437$	28.5028	1.36348
$438$	4.98486	0.238186
$439$	18.1319	0.865386	0.432693	−	0.901541i	$-0.357563\pi$
$439$	18.1319	0.865386	0.432693	+	0.901541i	$0.357563\pi$
$440$	0.742324	0.0353889
$441$	−5.63099	−0.268142
$442$	0.360735	0.0171584
$443$	−1.65471	−0.0786178	−0.0393089	−	0.999227i	$-0.512516\pi$
$443$	−1.65471	−0.0786178	−0.0393089	+	0.999227i	$0.512516\pi$
$444$	16.4622	0.781263
$445$	−1.46813	−0.0695961
$446$	−11.6833	−0.553222
$447$	−31.9329	−1.51037
$448$	−0.645165	−0.0304812
$449$	4.02871	0.190126	0.0950632	−	0.995471i	$-0.469695\pi$
$449$	4.02871	0.190126	0.0950632	+	0.995471i	$0.469695\pi$
$450$	−4.26046	−0.200840
$451$	38.7080	1.82269
$452$	2.96629	0.139523
$453$	12.3780	0.581571
$454$	14.3672	0.674284
$455$	−0.231384	−0.0108474
$456$	7.27906	0.340873
$457$	−9.73886	−0.455565	−0.227782	−	0.973712i	$-0.573147\pi$
$457$	−9.73886	−0.455565	−0.227782	+	0.973712i	$0.573147\pi$
$458$	−9.27281	−0.433290
$459$	0.578585	0.0270060
$460$	−1.05023	−0.0489672
$461$	7.82597	0.364492	0.182246	−	0.983253i	$-0.441663\pi$
$461$	7.82597	0.364492	0.182246	+	0.983253i	$0.441663\pi$
$462$	6.88414	0.320279
$463$	7.98347	0.371023	0.185512	−	0.982642i	$-0.440606\pi$
$463$	7.98347	0.371023	0.185512	+	0.982642i	$0.440606\pi$
$464$	−5.85445	−0.271786
$465$	−2.07743	−0.0963386
$466$	28.5096	1.32068
$467$	12.9077	0.597295	0.298648	−	0.954363i	$-0.403464\pi$
$467$	12.9077	0.597295	0.298648	+	0.954363i	$0.403464\pi$
$468$	2.24559	0.103802
$469$	6.96277	0.321511
$470$	−1.04216	−0.0480713
$471$	−17.1061	−0.788208
$472$	10.6054	0.488152
$473$	−53.2673	−2.44923
$474$	−2.55341	−0.117282
$475$	18.4669	0.847318
$476$	−0.0886421	−0.00406290
$477$	5.52061	0.252771
$478$	−1.92886	−0.0882238
$479$	−1.55841	−0.0712058	−0.0356029	−	0.999366i	$-0.511335\pi$
$479$	−1.55841	−0.0712058	−0.0356029	+	0.999366i	$0.511335\pi$
$480$	−0.268208	−0.0122419
$481$	−22.0130	−1.00371
$482$	−17.7798	−0.809849
$483$	−9.73958	−0.443166
$484$	18.5326	0.842390
$485$	−0.204005	−0.00926340
$486$	8.63973	0.391906
$487$	−9.52027	−0.431405	−0.215702	−	0.976459i	$-0.569204\pi$
$487$	−9.52027	−0.431405	−0.215702	+	0.976459i	$0.569204\pi$
$488$	10.8817	0.492589
$489$	26.9137	1.21708
$490$	−0.899325	−0.0406274
$491$	−21.3348	−0.962825	−0.481413	−	0.876494i	$-0.659876\pi$
$491$	−21.3348	−0.962825	−0.481413	+	0.876494i	$0.659876\pi$
$492$	−13.9855	−0.630516
$493$	−0.804369	−0.0362270
$494$	−9.73343	−0.437928
$495$	0.634898	0.0285366
$496$	7.74561	0.347788
$497$	−4.79852	−0.215243
$498$	33.9213	1.52005
$499$	8.70138	0.389527	0.194764	−	0.980850i	$-0.437606\pi$
$499$	8.70138	0.389527	0.194764	+	0.980850i	$0.437606\pi$
$500$	−1.36343	−0.0609743
$501$	44.8609	2.00424
$502$	−21.1878	−0.945656
$503$	39.1479	1.74552	0.872758	−	0.488152i	$-0.162329\pi$
$503$	39.1479	1.74552	0.872758	+	0.488152i	$0.162329\pi$
$504$	−0.551800	−0.0245791
$505$	−0.0231994	−0.00103236
$506$	−41.7822	−1.85745
$507$	11.9901	0.532499
$508$	8.58178	0.380755
$509$	−30.7906	−1.36477	−0.682384	−	0.730994i	$-0.739057\pi$
$509$	−30.7906	−1.36477	−0.682384	+	0.730994i	$0.739057\pi$
$510$	−0.0368502	−0.00163175
$511$	1.63793	0.0724578
$512$	1.00000	0.0441942
$513$	−15.6115	−0.689265
$514$	−27.9902	−1.23459
$515$	−1.22590	−0.0540195
$516$	19.2459	0.847254
$517$	−41.4613	−1.82347
$518$	5.40918	0.237666
$519$	26.9706	1.18388
$520$	0.358642	0.0157275
$521$	−21.1470	−0.926467	−0.463233	−	0.886236i	$-0.653311\pi$
$521$	−21.1470	−0.926467	−0.463233	+	0.886236i	$0.653311\pi$
$522$	−5.00722	−0.219160
$523$	−9.33978	−0.408400	−0.204200	−	0.978929i	$-0.565459\pi$
$523$	−9.33978	−0.408400	−0.204200	+	0.978929i	$0.565459\pi$
$524$	18.2274	0.796269
$525$	−6.31024	−0.275401
$526$	−23.4759	−1.02360
$527$	1.06420	0.0463574
$528$	−10.6704	−0.464368
$529$	36.1129	1.57013
$530$	0.881696	0.0382984
$531$	9.07061	0.393631
$532$	2.39176	0.103696
$533$	18.7012	0.810038
$534$	21.1033	0.913229
$535$	2.23148	0.0964752
$536$	−10.7922	−0.466153
$537$	−0.447943	−0.0193301
$538$	−19.9201	−0.858817
$539$	−35.7787	−1.54110
$540$	0.575229	0.0247539
$541$	−21.1622	−0.909836	−0.454918	−	0.890533i	$-0.650331\pi$
$541$	−21.1622	−0.909836	−0.454918	+	0.890533i	$0.650331\pi$
$542$	−12.7918	−0.549455
$543$	46.5315	1.99686
$544$	0.137394	0.00589074
$545$	−0.384222	−0.0164583
$546$	3.32597	0.142338
$547$	−43.1738	−1.84598	−0.922989	−	0.384825i	$-0.874262\pi$
$547$	−43.1738	−1.84598	−0.922989	+	0.384825i	$0.874262\pi$
$548$	2.28829	0.0977509
$549$	9.30691	0.397209
$550$	−27.0705	−1.15429
$551$	21.7037	0.924608
$552$	15.0963	0.642539
$553$	−0.839004	−0.0356781
$554$	18.8469	0.800729
$555$	2.24870	0.0954520
$556$	−4.39175	−0.186252
$557$	−2.08548	−0.0883647	−0.0441823	−	0.999023i	$-0.514068\pi$
$557$	−2.08548	−0.0883647	−0.0441823	+	0.999023i	$0.514068\pi$
$558$	6.62470	0.280446
$559$	−25.7353	−1.08849
$560$	−0.0881280	−0.00372409
$561$	−1.46605	−0.0618966
$562$	−12.9456	−0.546076
$563$	34.2456	1.44328	0.721640	−	0.692268i	$-0.243389\pi$
$563$	34.2456	1.44328	0.721640	+	0.692268i	$0.243389\pi$
$564$	14.9803	0.630784
$565$	0.405188	0.0170464
$566$	−17.4489	−0.733431
$567$	6.98994	0.293550
$568$	7.43766	0.312077
$569$	−16.2887	−0.682857	−0.341429	−	0.939908i	$-0.610911\pi$
$569$	−16.2887	−0.682857	−0.341429	+	0.939908i	$0.610911\pi$
$570$	0.994301	0.0416467
$571$	29.0341	1.21504	0.607519	−	0.794305i	$-0.292165\pi$
$571$	29.0341	1.21504	0.607519	+	0.794305i	$0.292165\pi$
$572$	14.2682	0.596584
$573$	38.1758	1.59482
$574$	−4.59538	−0.191807
$575$	38.2990	1.59718
$576$	0.855285	0.0356369
$577$	−24.9873	−1.04024	−0.520118	−	0.854094i	$-0.674112\pi$
$577$	−24.9873	−1.04024	−0.520118	+	0.854094i	$0.674112\pi$
$578$	−16.9811	−0.706322
$579$	18.4083	0.765021
$580$	−0.799704	−0.0332059
$581$	11.1459	0.462410
$582$	2.93242	0.121553
$583$	35.0773	1.45276
$584$	−2.53878	−0.105055
$585$	0.306741	0.0126822
$586$	−12.8542	−0.531002
$587$	−2.00171	−0.0826195	−0.0413098	−	0.999146i	$-0.513153\pi$
$587$	−2.00171	−0.0826195	−0.0413098	+	0.999146i	$0.513153\pi$
$588$	12.9271	0.533106
$589$	−28.7146	−1.18316
$590$	1.44867	0.0596407
$591$	29.2136	1.20169
$592$	−8.38418	−0.344587
$593$	−17.5875	−0.722232	−0.361116	−	0.932521i	$-0.617604\pi$
$593$	−17.5875	−0.722232	−0.361116	+	0.932521i	$0.617604\pi$
$594$	22.8849	0.938978
$595$	−0.0121083	−0.000496391 0
$596$	16.2633	0.666172
$597$	−34.5553	−1.41425
$598$	−20.1865	−0.825485
$599$	20.2843	0.828792	0.414396	−	0.910097i	$-0.363993\pi$
$599$	20.2843	0.828792	0.414396	+	0.910097i	$0.363993\pi$
$600$	9.78080	0.399300
$601$	−43.8527	−1.78879	−0.894394	−	0.447280i	$-0.852393\pi$
$601$	−43.8527	−1.78879	−0.894394	+	0.447280i	$0.852393\pi$
$602$	6.32385	0.257741
$603$	−9.23042	−0.375892
$604$	−6.30411	−0.256511
$605$	2.53150	0.102920
$606$	0.333474	0.0135465
$607$	−5.40321	−0.219310	−0.109655	−	0.993970i	$-0.534975\pi$
$607$	−5.40321	−0.219310	−0.109655	+	0.993970i	$0.534975\pi$
$608$	−3.70721	−0.150347
$609$	−7.41627	−0.300522
$610$	1.48641	0.0601829
$611$	−20.0314	−0.810383
$612$	0.117511	0.00475011
$613$	12.0452	0.486501	0.243250	−	0.969964i	$-0.421786\pi$
$613$	12.0452	0.486501	0.243250	+	0.969964i	$0.421786\pi$
$614$	25.2718	1.01989
$615$	−1.91039	−0.0770342
$616$	−3.50608	−0.141264
$617$	35.5498	1.43118	0.715590	−	0.698520i	$-0.246158\pi$
$617$	35.5498	1.43118	0.715590	+	0.698520i	$0.246158\pi$
$618$	17.6214	0.708836
$619$	−27.4798	−1.10450	−0.552252	−	0.833677i	$-0.686231\pi$
$619$	−27.4798	−1.10450	−0.552252	+	0.833677i	$0.686231\pi$
$620$	1.05803	0.0424915
$621$	−32.3772	−1.29925
$622$	8.91856	0.357602
$623$	6.93415	0.277811
$624$	−5.15522	−0.206374
$625$	24.7205	0.988819
$626$	−28.9112	−1.15552
$627$	39.5572	1.57976
$628$	8.71210	0.347651
$629$	−1.15194	−0.0459308
$630$	−0.0753745	−0.00300299
$631$	−16.5309	−0.658083	−0.329042	−	0.944315i	$-0.606726\pi$
$631$	−16.5309	−0.658083	−0.329042	+	0.944315i	$0.606726\pi$
$632$	1.30045	0.0517290
$633$	−48.5822	−1.93097
$634$	−14.6613	−0.582273
$635$	1.17225	0.0465193
$636$	−12.6737	−0.502546
$637$	−17.2859	−0.684894
$638$	−31.8154	−1.25958
$639$	6.36131	0.251650
$640$	0.136598	0.00539949
$641$	18.7497	0.740571	0.370285	−	0.928918i	$-0.379260\pi$
$641$	18.7497	0.740571	0.370285	+	0.928918i	$0.379260\pi$
$642$	−32.0758	−1.26593
$643$	22.6413	0.892884	0.446442	−	0.894812i	$-0.352691\pi$
$643$	22.6413	0.892884	0.446442	+	0.894812i	$0.352691\pi$
$644$	4.96035	0.195465
$645$	2.62894	0.103515
$646$	−0.509350	−0.0200401
$647$	7.68801	0.302247	0.151123	−	0.988515i	$-0.451711\pi$
$647$	7.68801	0.302247	0.151123	+	0.988515i	$0.451711\pi$
$648$	−10.8343	−0.425613
$649$	57.6337	2.26232
$650$	−13.0787	−0.512990
$651$	9.81194	0.384560
$652$	−13.7071	−0.536810
$653$	12.3860	0.484703	0.242351	−	0.970189i	$-0.422081\pi$
$653$	12.3860	0.484703	0.242351	+	0.970189i	$0.422081\pi$
$654$	5.52291	0.215963
$655$	2.48982	0.0972853
$656$	7.12279	0.278098
$657$	−2.17138	−0.0847135
$658$	4.92224	0.191889
$659$	47.4180	1.84714	0.923571	−	0.383427i	$-0.125256\pi$
$659$	47.4180	1.84714	0.923571	+	0.383427i	$0.125256\pi$
$660$	−1.45754	−0.0567348
$661$	−12.0229	−0.467638	−0.233819	−	0.972280i	$-0.575122\pi$
$661$	−12.0229	−0.467638	−0.233819	+	0.972280i	$0.575122\pi$
$662$	6.90707	0.268451
$663$	−0.708298	−0.0275080
$664$	−17.2760	−0.670441
$665$	0.326709	0.0126692
$666$	−7.17086	−0.277865
$667$	45.0119	1.74287
$668$	−22.8476	−0.883999
$669$	22.9401	0.886915
$670$	−1.47419	−0.0569530
$671$	59.1351	2.28289
$672$	1.26677	0.0488669
$673$	−35.4126	−1.36506	−0.682528	−	0.730860i	$-0.739119\pi$
$673$	−35.4126	−1.36506	−0.682528	+	0.730860i	$0.739119\pi$
$674$	18.4880	0.712132
$675$	−20.9770	−0.807407
$676$	−6.10653	−0.234867
$677$	28.9929	1.11429	0.557144	−	0.830416i	$-0.311897\pi$
$677$	28.9929	1.11429	0.557144	+	0.830416i	$0.311897\pi$
$678$	−5.82428	−0.223680
$679$	0.963539	0.0369772
$680$	0.0187677	0.000719710 0
$681$	−28.2098	−1.08100
$682$	42.0927	1.61181
$683$	−31.8393	−1.21830	−0.609148	−	0.793056i	$-0.708489\pi$
$683$	−31.8393	−1.21830	−0.609148	+	0.793056i	$0.708489\pi$
$684$	−3.17072	−0.121235
$685$	0.312575	0.0119429
$686$	8.76377	0.334602
$687$	18.2070	0.694642
$688$	−9.80190	−0.373694
$689$	16.9471	0.645632
$690$	2.06211	0.0785032
$691$	18.3672	0.698722	0.349361	−	0.936988i	$-0.386399\pi$
$691$	18.3672	0.698722	0.349361	+	0.936988i	$0.386399\pi$
$692$	−13.7361	−0.522167
$693$	−2.99870	−0.113911
$694$	8.90909	0.338184
$695$	−0.599902	−0.0227556
$696$	11.4951	0.435722
$697$	0.978632	0.0370683
$698$	6.31107	0.238878
$699$	−55.9782	−2.11729
$700$	3.21379	0.121470
$701$	42.7075	1.61304	0.806520	−	0.591207i	$-0.201348\pi$
$701$	42.7075	1.61304	0.806520	+	0.591207i	$0.201348\pi$
$702$	11.0565	0.417300
$703$	31.0819	1.17228
$704$	5.43439	0.204816
$705$	2.04627	0.0770670
$706$	3.99399	0.150316
$707$	0.109573	0.00412093
$708$	−20.8235	−0.782596
$709$	−51.1452	−1.92080	−0.960398	−	0.278630i	$-0.910120\pi$
$709$	−51.1452	−1.92080	−0.960398	+	0.278630i	$0.910120\pi$
$710$	1.01597	0.0381285
$711$	1.11225	0.0417127
$712$	−10.7479	−0.402793
$713$	−59.5520	−2.23024
$714$	0.174048	0.00651357
$715$	1.94900	0.0728886
$716$	0.228136	0.00852585
$717$	3.78729	0.141439
$718$	21.0913	0.787121
$719$	−15.9336	−0.594222	−0.297111	−	0.954843i	$-0.596023\pi$
$719$	−15.9336	−0.594222	−0.297111	+	0.954843i	$0.596023\pi$
$720$	0.116830	0.00435399
$721$	5.79005	0.215633
$722$	−5.25661	−0.195631
$723$	34.9105	1.29833
$724$	−23.6984	−0.880745
$725$	29.1630	1.08309
$726$	−36.3885	−1.35050
$727$	−45.3193	−1.68080	−0.840400	−	0.541967i	$-0.817680\pi$
$727$	−45.3193	−1.68080	−0.840400	+	0.541967i	$0.817680\pi$
$728$	−1.69391	−0.0627804
$729$	15.5390	0.575519
$730$	−0.346791	−0.0128353
$731$	−1.34673	−0.0498105
$732$	−21.3660	−0.789710
$733$	−32.3169	−1.19365	−0.596826	−	0.802371i	$-0.703572\pi$
$733$	−32.3169	−1.19365	−0.596826	+	0.802371i	$0.703572\pi$
$734$	7.10645	0.262304
$735$	1.76581	0.0651331
$736$	−7.68849	−0.283401
$737$	−58.6491	−2.16037
$738$	6.09201	0.224250
$739$	19.0615	0.701187	0.350594	−	0.936528i	$-0.385980\pi$
$739$	19.0615	0.701187	0.350594	+	0.936528i	$0.385980\pi$
$740$	−1.14526	−0.0421005
$741$	19.1115	0.702077
$742$	−4.16435	−0.152878
$743$	50.4184	1.84967	0.924836	−	0.380366i	$-0.124202\pi$
$743$	50.4184	1.84967	0.924836	+	0.380366i	$0.124202\pi$
$744$	−15.2084	−0.557567
$745$	2.22153	0.0813906
$746$	−20.1340	−0.737158
$747$	−14.7759	−0.540623
$748$	0.746655	0.0273004
$749$	−10.5395	−0.385106
$750$	2.67707	0.0977528
$751$	19.2912	0.703945	0.351973	−	0.936010i	$-0.385511\pi$
$751$	19.2912	0.703945	0.351973	+	0.936010i	$0.385511\pi$
$752$	−7.62943	−0.278217
$753$	41.6019	1.51606
$754$	−15.3711	−0.559783
$755$	−0.861125	−0.0313396
$756$	−2.71687	−0.0988116
$757$	−22.6521	−0.823303	−0.411651	−	0.911341i	$-0.635048\pi$
$757$	−22.6521	−0.823303	−0.411651	+	0.911341i	$0.635048\pi$
$758$	−15.6170	−0.567234
$759$	82.0389	2.97782
$760$	−0.506395	−0.0183689
$761$	−30.1840	−1.09417	−0.547084	−	0.837078i	$-0.684262\pi$
$761$	−30.1840	−1.09417	−0.547084	+	0.837078i	$0.684262\pi$
$762$	−16.8502	−0.610419
$763$	1.81472	0.0656974
$764$	−19.4428	−0.703417
$765$	0.0160518	0.000580352 0
$766$	−23.6516	−0.854567
$767$	27.8448	1.00542
$768$	−1.96349	−0.0708513
$769$	−36.8802	−1.32993	−0.664967	−	0.746872i	$-0.731555\pi$
$769$	−36.8802	−1.32993	−0.664967	+	0.746872i	$0.731555\pi$
$770$	−0.478922	−0.0172591
$771$	54.9584	1.97928
$772$	−9.37529	−0.337424
$773$	5.77620	0.207755	0.103878	−	0.994590i	$-0.466875\pi$
$773$	5.77620	0.207755	0.103878	+	0.994590i	$0.466875\pi$
$774$	−8.38341	−0.301336
$775$	−38.5835	−1.38596
$776$	−1.49348	−0.0536127
$777$	−10.6209	−0.381021
$778$	−5.10491	−0.183020
$779$	−26.4057	−0.946081
$780$	−0.704190	−0.0252141
$781$	40.4191	1.44631
$782$	−1.05636	−0.0377752
$783$	−24.6538	−0.881055
$784$	−6.58376	−0.235134
$785$	1.19005	0.0424748
$786$	−35.7893	−1.27656
$787$	6.57884	0.234510	0.117255	−	0.993102i	$-0.462590\pi$
$787$	6.57884	0.234510	0.117255	+	0.993102i	$0.462590\pi$
$788$	−14.8784	−0.530021
$789$	46.0946	1.64101
$790$	0.177638	0.00632007
$791$	−1.91375	−0.0680451
$792$	4.64795	0.165158
$793$	28.5702	1.01456
$794$	39.6218	1.40613
$795$	−1.73120	−0.0613993
$796$	17.5989	0.623778
$797$	50.0108	1.77147	0.885737	−	0.464187i	$-0.153653\pi$
$797$	50.0108	1.77147	0.885737	+	0.464187i	$0.153653\pi$
$798$	−4.69620	−0.166244
$799$	−1.04824	−0.0370841
$800$	−4.98134	−0.176117
$801$	−9.19248	−0.324800
$802$	−25.0412	−0.884234
$803$	−13.7967	−0.486875
$804$	21.1904	0.747328
$805$	0.677571	0.0238812
$806$	20.3364	0.716320
$807$	39.1129	1.37684
$808$	−0.169838	−0.00597487
$809$	22.4710	0.790037	0.395018	−	0.918673i	$-0.370738\pi$
$809$	22.4710	0.790037	0.395018	+	0.918673i	$0.370738\pi$
$810$	−1.47994	−0.0519999
$811$	−29.1457	−1.02345	−0.511723	−	0.859151i	$-0.670992\pi$
$811$	−29.1457	−1.02345	−0.511723	+	0.859151i	$0.670992\pi$
$812$	3.77709	0.132550
$813$	25.1165	0.880875
$814$	−45.5629	−1.59698
$815$	−1.87235	−0.0655856
$816$	−0.269772	−0.00944392
$817$	36.3377	1.27129
$818$	30.8092	1.07722
$819$	−1.44877	−0.0506243
$820$	0.972956	0.0339771
$821$	−20.1165	−0.702072	−0.351036	−	0.936362i	$-0.614171\pi$
$821$	−20.1165	−0.702072	−0.351036	+	0.936362i	$0.614171\pi$
$822$	−4.49303	−0.156712
$823$	20.9550	0.730445	0.365223	−	0.930920i	$-0.380993\pi$
$823$	20.9550	0.730445	0.365223	+	0.930920i	$0.380993\pi$
$824$	−8.97453	−0.312642
$825$	53.1527	1.85054
$826$	−6.84222	−0.238071
$827$	33.8858	1.17832	0.589162	−	0.808015i	$-0.299458\pi$
$827$	33.8858	1.17832	0.589162	+	0.808015i	$0.299458\pi$
$828$	−6.57585	−0.228527
$829$	10.9601	0.380659	0.190329	−	0.981720i	$-0.439044\pi$
$829$	10.9601	0.380659	0.190329	+	0.981720i	$0.439044\pi$
$830$	−2.35986	−0.0819121
$831$	−37.0057	−1.28371
$832$	2.62554	0.0910243
$833$	−0.904572	−0.0313416
$834$	8.62314	0.298595
$835$	−3.12092	−0.108004
$836$	−20.1464	−0.696778
$837$	32.6177	1.12743
$838$	23.2766	0.804076
$839$	−2.45946	−0.0849100	−0.0424550	−	0.999098i	$-0.513518\pi$
$839$	−2.45946	−0.0849100	−0.0424550	+	0.999098i	$0.513518\pi$
$840$	0.173038	0.00597039
$841$	5.27461	0.181883
$842$	−16.6580	−0.574071
$843$	25.4185	0.875459
$844$	24.7428	0.851682
$845$	−0.834137	−0.0286952
$846$	−6.52533	−0.224346
$847$	−11.9566	−0.410833
$848$	6.45470	0.221655
$849$	34.2606	1.17582
$850$	−0.684408	−0.0234750
$851$	64.4616	2.20972
$852$	−14.6037	−0.500316
$853$	−20.7983	−0.712120	−0.356060	−	0.934463i	$-0.615880\pi$
$853$	−20.7983	−0.712120	−0.356060	+	0.934463i	$0.615880\pi$
$854$	−7.02047	−0.240235
$855$	−0.433112	−0.0148121
$856$	16.3362	0.558358
$857$	−45.4291	−1.55183	−0.775914	−	0.630839i	$-0.782711\pi$
$857$	−45.4291	−1.55183	−0.775914	+	0.630839i	$0.782711\pi$
$858$	−28.0155	−0.956432
$859$	−2.75793	−0.0940993	−0.0470496	−	0.998893i	$-0.514982\pi$
$859$	−2.75793	−0.0940993	−0.0470496	+	0.998893i	$0.514982\pi$
$860$	−1.33891	−0.0456566
$861$	9.02297	0.307502
$862$	3.10680	0.105818
$863$	−38.5720	−1.31301	−0.656503	−	0.754323i	$-0.727965\pi$
$863$	−38.5720	−1.31301	−0.656503	+	0.754323i	$0.727965\pi$
$864$	4.21112	0.143265
$865$	−1.87631	−0.0637965
$866$	−17.9951	−0.611498
$867$	33.3422	1.13236
$868$	−4.99720	−0.169616
$869$	7.06714	0.239736
$870$	1.57021	0.0532350
$871$	−28.3354	−0.960110
$872$	−2.81281	−0.0952536
$873$	−1.27735	−0.0432317
$874$	28.5028	0.964123
$875$	0.879635	0.0297371
$876$	4.98486	0.168423
$877$	25.9340	0.875730	0.437865	−	0.899041i	$-0.355735\pi$
$877$	25.9340	0.875730	0.437865	+	0.899041i	$0.355735\pi$
$878$	18.1319	0.611920
$879$	25.2390	0.851292
$880$	0.742324	0.0250237
$881$	32.2921	1.08795	0.543974	−	0.839102i	$-0.316919\pi$
$881$	32.2921	1.08795	0.543974	+	0.839102i	$0.316919\pi$
$882$	−5.63099	−0.189605
$883$	−47.8695	−1.61094	−0.805468	−	0.592639i	$-0.798086\pi$
$883$	−47.8695	−1.61094	−0.805468	+	0.592639i	$0.798086\pi$
$884$	0.360735	0.0121328
$885$	−2.84444	−0.0956148
$886$	−1.65471	−0.0555912
$887$	40.6247	1.36405	0.682023	−	0.731331i	$-0.261101\pi$
$887$	40.6247	1.36405	0.682023	+	0.731331i	$0.261101\pi$
$888$	16.4622	0.552436
$889$	−5.53666	−0.185694
$890$	−1.46813	−0.0492118
$891$	−58.8780	−1.97249
$892$	−11.6833	−0.391187
$893$	28.2839	0.946484
$894$	−31.9329	−1.06799
$895$	0.0311628	0.00104166
$896$	−0.645165	−0.0215535
$897$	39.6359	1.32340
$898$	4.02871	0.134440
$899$	−45.3463	−1.51238
$900$	−4.26046	−0.142015
$901$	0.886839	0.0295449
$902$	38.7080	1.28884
$903$	−12.4168	−0.413205
$904$	2.96629	0.0986574
$905$	−3.23714	−0.107606
$906$	12.3780	0.411233
$907$	13.6558	0.453433	0.226717	−	0.973961i	$-0.427201\pi$
$907$	13.6558	0.453433	0.226717	+	0.973961i	$0.427201\pi$
$908$	14.3672	0.476791
$909$	−0.145260	−0.00481796
$910$	−0.231384	−0.00767030
$911$	−47.1056	−1.56068	−0.780339	−	0.625357i	$-0.784953\pi$
$911$	−47.1056	−1.56068	−0.780339	+	0.625357i	$0.784953\pi$
$912$	7.27906	0.241034
$913$	−93.8848	−3.10713
$914$	−9.73886	−0.322133
$915$	−2.91854	−0.0964840
$916$	−9.27281	−0.306382
$917$	−11.7597	−0.388340
$918$	0.578585	0.0190961
$919$	25.9098	0.854685	0.427342	−	0.904090i	$-0.359450\pi$
$919$	25.9098	0.854685	0.427342	+	0.904090i	$0.359450\pi$
$920$	−1.05023	−0.0346250
$921$	−49.6208	−1.63506
$922$	7.82597	0.257735
$923$	19.5279	0.642768
$924$	6.88414	0.226472
$925$	41.7644	1.37321
$926$	7.98347	0.262353
$927$	−7.67578	−0.252106
$928$	−5.85445	−0.192182
$929$	−26.8377	−0.880515	−0.440257	−	0.897872i	$-0.645113\pi$
$929$	−26.8377	−0.880515	−0.440257	+	0.897872i	$0.645113\pi$
$930$	−2.07743	−0.0681217
$931$	24.4074	0.799919
$932$	28.5096	0.933861
$933$	−17.5115	−0.573300
$934$	12.9077	0.422352
$935$	0.101991	0.00333547
$936$	2.24559	0.0733993
$937$	43.3481	1.41612	0.708060	−	0.706153i	$-0.249571\pi$
$937$	43.3481	1.41612	0.708060	+	0.706153i	$0.249571\pi$
$938$	6.96277	0.227342
$939$	56.7668	1.85251
$940$	−1.04216	−0.0339915
$941$	−57.9078	−1.88774	−0.943871	−	0.330315i	$-0.892845\pi$
$941$	−57.9078	−1.88774	−0.943871	+	0.330315i	$0.892845\pi$
$942$	−17.1061	−0.557347
$943$	−54.7635	−1.78335
$944$	10.6054	0.345175
$945$	−0.371118	−0.0120725
$946$	−53.2673	−1.73187
$947$	−30.9144	−1.00458	−0.502292	−	0.864698i	$-0.667510\pi$
$947$	−30.9144	−1.00458	−0.502292	+	0.864698i	$0.667510\pi$
$948$	−2.55341	−0.0829310
$949$	−6.66567	−0.216377
$950$	18.4669	0.599144
$951$	28.7872	0.933489
$952$	−0.0886421	−0.00287291
$953$	41.9830	1.35996	0.679982	−	0.733229i	$-0.261988\pi$
$953$	41.9830	1.35996	0.679982	+	0.733229i	$0.261988\pi$
$954$	5.52061	0.178736
$955$	−2.65584	−0.0859411
$956$	−1.92886	−0.0623837
$957$	62.4691	2.01934
$958$	−1.55841	−0.0503501
$959$	−1.47633	−0.0476730
$960$	−0.268208	−0.00865636
$961$	28.9945	0.935306
$962$	−22.0130	−0.709728
$963$	13.9721	0.450243
$964$	−17.7798	−0.572650
$965$	−1.28064	−0.0412253
$966$	−9.73958	−0.313366
$967$	−7.22172	−0.232235	−0.116117	−	0.993235i	$-0.537045\pi$
$967$	−7.22172	−0.232235	−0.116117	+	0.993235i	$0.537045\pi$
$968$	18.5326	0.595660
$969$	1.00010	0.0321279
$970$	−0.204005	−0.00655021
$971$	−8.26287	−0.265168	−0.132584	−	0.991172i	$-0.542327\pi$
$971$	−8.26287	−0.265168	−0.132584	+	0.991172i	$0.542327\pi$
$972$	8.63973	0.277120
$973$	2.83340	0.0908348
$974$	−9.52027	−0.305049
$975$	25.6799	0.822415
$976$	10.8817	0.348313
$977$	−8.27305	−0.264678	−0.132339	−	0.991204i	$-0.542249\pi$
$977$	−8.27305	−0.264678	−0.132339	+	0.991204i	$0.542249\pi$
$978$	26.9137	0.860604
$979$	−58.4080	−1.86673
$980$	−0.899325	−0.0287279
$981$	−2.40575	−0.0768097
$982$	−21.3348	−0.680820
$983$	−16.8305	−0.536810	−0.268405	−	0.963306i	$-0.586497\pi$
$983$	−16.8305	−0.536810	−0.268405	+	0.963306i	$0.586497\pi$
$984$	−13.9855	−0.445842
$985$	−2.03235	−0.0647562
$986$	−0.804369	−0.0256163
$987$	−9.66477	−0.307633
$988$	−9.73343	−0.309662
$989$	75.3618	2.39637
$990$	0.634898	0.0201784
$991$	27.6418	0.878071	0.439036	−	0.898470i	$-0.355320\pi$
$991$	27.6418	0.878071	0.439036	+	0.898470i	$0.355320\pi$
$992$	7.74561	0.245923
$993$	−13.5620	−0.430376
$994$	−4.79852	−0.152200
$995$	2.40397	0.0762110
$996$	33.9213	1.07484
$997$	23.4908	0.743960	0.371980	−	0.928241i	$-0.378679\pi$
$997$	23.4908	0.743960	0.371980	+	0.928241i	$0.378679\pi$
$998$	8.70138	0.275437
$999$	−35.3068	−1.11706

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.d.1.20	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.d.1.20	✓	69	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.96349 q^{3} +1.00000 q^{4} +0.136598 q^{5} -1.96349 q^{6} -0.645165 q^{7} +1.00000 q^{8} +0.855285 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.96349 q^{3} +1.00000 q^{4} +0.136598 q^{5} -1.96349 q^{6} -0.645165 q^{7} +1.00000 q^{8} +0.855285 q^{9} +0.136598 q^{10} +5.43439 q^{11} -1.96349 q^{12} +2.62554 q^{13} -0.645165 q^{14} -0.268208 q^{15} +1.00000 q^{16} +0.137394 q^{17} +0.855285 q^{18} -3.70721 q^{19} +0.136598 q^{20} +1.26677 q^{21} +5.43439 q^{22} -7.68849 q^{23} -1.96349 q^{24} -4.98134 q^{25} +2.62554 q^{26} +4.21112 q^{27} -0.645165 q^{28} -5.85445 q^{29} -0.268208 q^{30} +7.74561 q^{31} +1.00000 q^{32} -10.6704 q^{33} +0.137394 q^{34} -0.0881280 q^{35} +0.855285 q^{36} -8.38418 q^{37} -3.70721 q^{38} -5.15522 q^{39} +0.136598 q^{40} +7.12279 q^{41} +1.26677 q^{42} -9.80190 q^{43} +5.43439 q^{44} +0.116830 q^{45} -7.68849 q^{46} -7.62943 q^{47} -1.96349 q^{48} -6.58376 q^{49} -4.98134 q^{50} -0.269772 q^{51} +2.62554 q^{52} +6.45470 q^{53} +4.21112 q^{54} +0.742324 q^{55} -0.645165 q^{56} +7.27906 q^{57} -5.85445 q^{58} +10.6054 q^{59} -0.268208 q^{60} +10.8817 q^{61} +7.74561 q^{62} -0.551800 q^{63} +1.00000 q^{64} +0.358642 q^{65} -10.6704 q^{66} -10.7922 q^{67} +0.137394 q^{68} +15.0963 q^{69} -0.0881280 q^{70} +7.43766 q^{71} +0.855285 q^{72} -2.53878 q^{73} -8.38418 q^{74} +9.78080 q^{75} -3.70721 q^{76} -3.50608 q^{77} -5.15522 q^{78} +1.30045 q^{79} +0.136598 q^{80} -10.8343 q^{81} +7.12279 q^{82} -17.2760 q^{83} +1.26677 q^{84} +0.0187677 q^{85} -9.80190 q^{86} +11.4951 q^{87} +5.43439 q^{88} -10.7479 q^{89} +0.116830 q^{90} -1.69391 q^{91} -7.68849 q^{92} -15.2084 q^{93} -7.62943 q^{94} -0.506395 q^{95} -1.96349 q^{96} -1.49348 q^{97} -6.58376 q^{98} +4.64795 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * q^99

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